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Properties

Label 2-15e2-9.7-c3-0-21
Degree $2$
Conductor $225$
Sign $0.230 - 0.973i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0874 + 0.151i)2-s + (−5.19 − 0.151i)3-s + (3.98 + 6.90i)4-s + (0.477 − 0.773i)6-s + (4.23 − 7.32i)7-s − 2.79·8-s + (26.9 + 1.57i)9-s + (15.7 − 27.2i)11-s + (−19.6 − 36.4i)12-s + (13.4 + 23.2i)13-s + (0.740 + 1.28i)14-s + (−31.6 + 54.7i)16-s + 44.3·17-s + (−2.59 + 3.94i)18-s − 90.2·19-s + ⋯
L(s)  = 1  + (−0.0309 + 0.0535i)2-s + (−0.999 − 0.0291i)3-s + (0.498 + 0.862i)4-s + (0.0324 − 0.0526i)6-s + (0.228 − 0.395i)7-s − 0.123·8-s + (0.998 + 0.0583i)9-s + (0.431 − 0.747i)11-s + (−0.472 − 0.876i)12-s + (0.286 + 0.496i)13-s + (0.0141 + 0.0244i)14-s + (−0.494 + 0.856i)16-s + 0.632·17-s + (−0.0340 + 0.0516i)18-s − 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.230 - 0.973i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07124 + 0.846892i\)
\(L(\frac12)\) \(\approx\) \(1.07124 + 0.846892i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.19 + 0.151i)T \)
5 \( 1 \)
good2 \( 1 + (0.0874 - 0.151i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-4.23 + 7.32i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-15.7 + 27.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-13.4 - 23.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 44.3T + 4.91e3T^{2} \)
19 \( 1 + 90.2T + 6.85e3T^{2} \)
23 \( 1 + (-97.1 - 168. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (1.87 - 3.24i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-125. - 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 62.2T + 5.06e4T^{2} \)
41 \( 1 + (-102. - 176. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (263. - 456. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-77.8 + 134. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 141.T + 1.48e5T^{2} \)
59 \( 1 + (-246. - 427. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-379. + 657. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (271. + 470. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 928.T + 3.57e5T^{2} \)
73 \( 1 + 608.T + 3.89e5T^{2} \)
79 \( 1 + (307. - 532. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-537. + 931. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (166. - 288. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.74706517778169875781347295534, −11.30323640567033664601933302677, −10.36959948631640223551928737336, −8.988984829570208603333565378899, −7.83751172151664660191339070166, −6.86472458020211558190306071324, −6.02249793957263667847471635891, −4.56333671952263543146822706293, −3.36649409898193014747705980746, −1.35366019637030434970450662021, 0.72109963794307047925168590818, 2.19220875340935021495674444448, 4.37358891304181179559579243127, 5.46187454079219452452570561736, 6.32925370399159828104795243861, 7.22266672207155051378442563620, 8.769654247784271545117953265885, 10.06891038109964686982753776423, 10.55832000037154243041026811908, 11.58172169123782392122855727048

Graph of the $Z$-function along the critical line